import Mathlib import OAI.Geometry.Cannon.Boundary.PolynomialPacking import OAI.Geometry.Cannon.Modulus.Clip namespace OAI section section section noncomputable section section section section section section section section section section section section section open Set MeasureTheory open scoped BigOperators open Set unitInterval namespace CannonAnnular open Metric noncomputable def pathCellIndices {Y β : Type*} [PseudoMetricSpace Y] [Fintype β] (ξ : β → Y) (s : ℝ) {u v : Y} (p : Path u v) : Finset β := by classical exact Finset.univ.filter (fun j => ∃ t, dist (p t) (ξ j) < 2*s) def MacroscopicAdmissible {Y β : Type*} [PseudoMetricSpace Y] [Fintype β] (ξ : β → Y) (s d₀ : ℝ) (ω : β → ℝ) : Prop := (∀ j, 0 ≤ ω j) ∧ ∀ (u v : Y) (p : Path u v), d₀ ≤ diam (Set.range p) → 1 ≤ ∑ j ∈ pathCellIndices ξ s p, ω j lemma exists_outer_point {X : Type*} [PseudoMetricSpace X] (o : X) (R : ℝ) (hR : 0 ≤ R) {K : Set X} (hdiam : 4*R < diam K) : ∃ y ∈ K, 2*R ≤ dist y o := by by_contra hn push Not at hn have hh : diam K ≤ 4*R := by apply diam_le_of_forall_dist_le (by positivity) intro x hx y hy have hh := dist_triangle x o y rw [dist_comm o y] at hh linarith [hn x hx, hn y hy] exact (not_lt_of_ge hh) hdiam theorem annular_interval_cover {X Y α β : Type*} [MetricSpace X] [MetricSpace Y] [LocallyPathConnectedSpace X] [Fintype α] [Fintype β] (q : α → X) (ξ : β → Y) (h : X → Y) (hh : Continuous h) (f : X → ℝ) (L U : α → ℝ) (o : X) (R r C d₀ a b : ℝ) (hR : 0 < R) (hr : 0 < r) (hrR : r ≤ R/2) (hC : 0 ≤ C) (hnet : ∀ y, R ≤ dist y o → dist y o ≤ 2*R → ∃ i, dist (q i) y ≤ r) (hzoom : ∀ u v, dist u o ≤ 3*R → dist v o ≤ 3*R → dist (h u) (h v) ≤ C/R * dist u v) (hlarge : ∀ u v, dist u o ≤ 2*R → dist v o ≤ 2*R → R ≤ dist u v → d₀ ≤ dist (h u) (h v)) (hrange : ∀ i z, dist z (q i) ≤ 2*r → f z ∈ Icc (L i) (U i)) (ω : β → ℝ) (hω : MacroscopicAdmissible ξ (r/R) d₀ ω) (Clevel : ℝ → Set X) (hlevel : ∀ t ∈ Ioo a b, IsPreconnected (Clevel t) ∧ 4*R < diam (Clevel t) ∧ (∃ x ∈ Clevel t, dist x o ≤ R/2) ∧ ∀ z ∈ Clevel t, f z = t) : ∀ t ∈ Ioo a b, 1 ≤ ∑ i, (Icc (L i) (U i)).indicator (fun _ : ℝ => ∑ j ∈ Finset.univ.filter (fun j => dist (ξ j) (h (q i)) ≤ (C+2)*(r/R)), ω j) t := by classical intro t ht obtain ⟨hc, hd, ⟨x, hx, hxr⟩, hf⟩ := hlevel t ht obtain ⟨y, hy, hyr⟩ := exists_outer_point o R hR.le hd let V : Set X := ⋃ z ∈ Clevel t, ball z (r/4) have hV : IsOpen V := isOpen_iUnion (fun z => isOpen_iUnion (fun _ => isOpen_ball)) have hCV : Clevel t ⊆ V := by intro z hz exact mem_iUnion.mpr ⟨z, mem_iUnion.mpr ⟨hz, by simp [hr]⟩⟩ obtain ⟨u, v, p, hpV, hu, hv, huv, hp⟩ := annular_path_near_continuum o R (2*R) (by linarith) (Clevel t) V hc hV hCV hx hy (by linarith) hyr let p' : Path (h u) (h v) := p.map hh have hmacro : d₀ ≤ diam (Set.range p') := by have hsep := hlarge u v (by linarith) hv.le (by linarith) apply hsep.trans exact dist_le_diam_of_mem (isCompact_range p'.continuous).isBounded ⟨0, p'.source⟩ ⟨1, p'.target⟩ have hadm := hω.2 (h u) (h v) p' hmacro apply interval_cover_of_charging _ ω L U hω.1 t (pathCellIndices ξ (r/R) p') hadm intro j hj obtain ⟨s, hs⟩ := (Finset.mem_filter.mp hj).2 change dist (h (p s)) (ξ j) < 2*(r/R) at hs have hsV := hpV (Set.mem_range_self s) obtain ⟨z, hz⟩ := mem_iUnion.mp hsV obtain ⟨hzC, hzs⟩ := mem_iUnion.mp hz have hnear : dist z (p s) ≤ r/4 := by rw [mem_ball, dist_comm] at hzs exact hzs.le obtain ⟨i, _, _, hit, hij⟩ := annular_net_charge q h f L U o (p s) z (ξ j) R r C t hR hr.le hrR hC (hp s) (hnet (p s) (hp s).1 (hp s).2) hzoom (by simpa only [dist_comm] using hs.le) hnear (hf z hzC) hrange exact ⟨i, hit, Finset.mem_filter.mpr ⟨Finset.mem_univ _, hij⟩⟩ end CannonAnnular namespace CannonAnnular open Metric theorem annular_comparison {X Y α β γ : Type*} [MetricSpace X] [MetricSpace Y] [LocallyPathConnectedSpace X] [Fintype α] [Fintype β] [Fintype γ] (q : α → X) (ξ : β → Y) (p : γ → X) (h : X → Y) (hh : Continuous h) (f : X → ℝ) (L U : α → ℝ) (o : X) (R r C d d₀ a b : ℝ) (hR : 0 < R) (hr : 0 < r) (hrR : r ≤ R/2) (hC : 0 ≤ C) (hd : d ≤ R/4) (hab : a ≤ b) (hLU : ∀ i, L i ≤ U i) (hq : ∀ i, R/2 ≤ dist (q i) o ∧ dist (q i) o ≤ 3*R) (hnet : ∀ y, R ≤ dist y o → dist y o ≤ 2*R → ∃ i, dist (q i) y ≤ r) (hzoom : ∀ u v, dist u o ≤ 3*R → dist v o ≤ 3*R → dist (h u) (h v) ≤ C/R * dist u v) (hlarge : ∀ u v, dist u o ≤ 2*R → dist v o ≤ 2*R → R ≤ dist u v → d₀ ≤ dist (h u) (h v)) (hrange : ∀ i z, dist z (q i) ≤ 2*r → f z ∈ Icc (L i) (U i)) (ω : β → ℝ) (hω : MacroscopicAdmissible ξ (r/R) d₀ ω) (ρ : γ → ℝ) (hρ : ∀ j, 0 ≤ ρ j) (hlocal : ∀ i, U i - L i ≤ ∑ j ∈ Finset.univ.filter (fun j => dist (p j) (q i) ≤ d), ρ j) (Clevel : ℝ → Set X) (hlevel : ∀ t ∈ Ioo a b, IsPreconnected (Clevel t) ∧ 4*R < diam (Clevel t) ∧ (∃ x ∈ Clevel t, dist x o ≤ R/2) ∧ ∀ z ∈ Clevel t, f z = t) (kr kc lr lc : ℕ) (hTrow : ∀ i, (Finset.univ.filter (fun j => dist (ξ j) (h (q i)) ≤ (C+2)*(r/R))).card ≤ kr) (hTcol : ∀ j, (Finset.univ.filter (fun i => dist (ξ j) (h (q i)) ≤ (C+2)*(r/R))).card ≤ kc) (hSrow : ∀ i, (Finset.univ.filter (fun j => dist (p j) (q i) ≤ d)).card ≤ lr) (hScol : ∀ j, (Finset.univ.filter (fun i => dist (p j) (q i) ≤ d)).card ≤ lc) : (b-a)^2 ≤ ((kr:ℝ)*kc*lr*lc) * (∑ j, (ω j)^2) * (∑ j ∈ Finset.univ.filter (fun j => R/4 ≤ dist (p j) o ∧ dist (p j) o ≤ 4*R), (ρ j)^2) := by classical let T (i : α) := Finset.univ.filter (fun j => dist (ξ j) (h (q i)) ≤ (C+2)*(r/R)) let S (i : α) := Finset.univ.filter (fun j => dist (p j) (q i) ≤ d) let A := Finset.univ.filter (fun j => R/4 ≤ dist (p j) o ∧ dist (p j) o ≤ 4*R) let w (j : γ) := if j ∈ A then ρ j else 0 have hn : ∀ j, 0 ≤ w j := by intro j dsimp [w] split_ifs · exact hρ j · exact le_rfl have hloc : ∀ i, U i - L i ≤ ∑ j ∈ S i, w j := by intro i apply (hlocal i).trans_eq apply Finset.sum_congr rfl intro j hj have hja : j ∈ A := Finset.mem_filter.mpr ⟨Finset.mem_univ _, charged_center_in_annulus o (q i) (p j) R d hR.le (hq i) (Finset.mem_filter.mp hj).2 hd⟩ simp [w, hja] have hcover := annular_interval_cover q ξ h hh f L U o R r C d₀ a b hR hr hrR hC hnet hzoom hlarge hrange ω hω Clevel hlevel have henergy := annular_energy_transfer T S ω w L U a b hab hω.1 hn hLU hloc kr kc lr lc hTrow (by simpa [T] using hTcol) hSrow (by simpa [S] using hScol) hcover have hwE : (∑ j, (w j)^2) = ∑ j ∈ A, (ρ j)^2 := by simp [w] rw [hwE] at henergy exact henergy end CannonAnnular open Set Metric namespace CannonAnnular open CannonPacking Metric theorem exists_packed_annular_constant {X : Type*} [MetricSpace X] [LocallyPathConnectedSpace X] (Q : ℕ) (CP C c A : ℝ) (hpack : PolynomialPacking (X := X) Q CP) (hCP : 0 ≤ CP) (hC : 0 ≤ C) (hc : 0 < c) (hA : 0 < A) : ∃ E : ℝ, 0 ≤ E ∧ ∀ {α β γ : Type*} [Fintype α] [Fintype β] [Fintype γ] (q : α → X) (ξ : β → X) (p : γ → X) (h : X → X), Continuous h → ∀ (f : X → ℝ) (L U : α → ℝ) (o : X) (R r d₀ a b : ℝ), 0 < R → 0 < r → r ≤ R/2 → A*r ≤ R/4 → max (C+2) (max c 1)*(r/R) ≤ 1 → max A 1*r ≤ 1 → a ≤ b → (∀ i, L i ≤ U i) → (∀ i, R/2 ≤ dist (q i) o ∧ dist (q i) o ≤ 3*R) → (∀ i j, i ≠ j → r ≤ dist (q i) (q j)) → (∀ i j, i ≠ j → r/R ≤ dist (ξ i) (ξ j)) → (∀ i j, i ≠ j → r ≤ dist (p i) (p j)) → (∀ y, R ≤ dist y o → dist y o ≤ 2*R → ∃ i, dist (q i) y ≤ r) → (∀ u v, dist u o ≤ 3*R → dist v o ≤ 3*R → dist (h u) (h v) ≤ C/R * dist u v) → (∀ u v, dist u o ≤ 3*R → dist v o ≤ 3*R → c/R * dist u v ≤ dist (h u) (h v)) → (∀ u v, dist u o ≤ 2*R → dist v o ≤ 2*R → R ≤ dist u v → d₀ ≤ dist (h u) (h v)) → (∀ i z, dist z (q i) ≤ 2*r → f z ∈ Icc (L i) (U i)) → ∀ (ω : β → ℝ), MacroscopicAdmissible ξ (r/R) d₀ ω → ∀ (ρ : γ → ℝ), (∀ j, 0 ≤ ρ j) → (∀ i, U i-L i ≤ ∑ j ∈ Finset.univ.filter (fun j => dist (p j) (q i) ≤ A*r), ρ j) → ∀ (Clevel : ℝ → Set X), (∀ t ∈ Ioo a b, IsPreconnected (Clevel t) ∧ 4*R < diam (Clevel t) ∧ (∃ x ∈ Clevel t, dist x o ≤ R/2) ∧ ∀ z ∈ Clevel t, f z = t) → (b-a)^2 ≤ E*(∑ j, (ω j)^2)* (∑ j ∈ Finset.univ.filter (fun j => R/4 ≤ dist (p j) o ∧ dist (p j) o ≤ 4*R), (ρ j)^2) := by classical have hC2 : 0 < C+2 := by linarith obtain ⟨kr, hkr⟩ := exists_overlap_bound Q CP (C+2) 1 hpack hCP hC2 zero_lt_one obtain ⟨kc, hkc⟩ := exists_overlap_bound Q CP (C+2) c hpack hCP hC2 hc obtain ⟨lr, hlr⟩ := exists_overlap_bound Q CP A 1 hpack hCP hA zero_lt_one obtain ⟨lc, hlc⟩ := exists_overlap_bound Q CP A 1 hpack hCP hA zero_lt_one refine ⟨(kr:ℝ)*kc*lr*lc, by positivity, ?_⟩ intro α β γ _ _ _ q ξ p h hh f L U o R r d₀ a b hR hr hrR hAr hscaleT hscaleS hab hLU hq hqsep hξsep hpsep hnet hzoom hlow hlarge hrange ω hω ρ hρ hlocal Clevel hlevel have hrdiv : 0 < r/R := div_pos hr hR have hscale1 : max (C+2) 1*(r/R) ≤ 1 := (mul_le_mul_of_nonneg_right (max_le_max le_rfl (le_max_right ..)) hrdiv.le).trans hscaleT have hscalec : max (C+2) c*(r/R) ≤ 1 := (mul_le_mul_of_nonneg_right (max_le_max le_rfl (le_max_left ..)) hrdiv.le).trans hscaleT have hrow (i : α) := hkr (ι := β) ξ Finset.univ (h (q i)) (r/R) hrdiv hscale1 (by simpa using hξsep) have himsep (i j : α) (hne : i ≠ j) : c*(r/R) ≤ dist (h (q i)) (h (q j)) := by calc c*(r/R) = c/R*r := by ring _ ≤ c/R*dist (q i) (q j) := mul_le_mul_of_nonneg_left (hqsep i j hne) (div_nonneg hc.le hR.le) _ ≤ dist (h (q i)) (h (q j)) := hlow _ _ (hq i).2 (hq j).2 have hcol (j : β) := hkc (ι := α) (fun i => h (q i)) Finset.univ (ξ j) (r/R) hrdiv hscalec (by simpa using himsep) have hsrow (i : α) := hlr (ι := γ) p Finset.univ (q i) r hr hscaleS (by simpa using hpsep) have hscol (j : γ) := hlc (ι := α) q Finset.univ (p j) r hr hscaleS (by simpa using hqsep) exact annular_comparison q ξ p h hh f L U o R r C (A*r) d₀ a b hR hr hrR hC hAr hab hLU hq hnet hzoom hlarge hrange ω hω ρ hρ hlocal Clevel hlevel kr kc lr lc (by simpa only [dist_comm] using hrow) hcol (by simpa only [one_mul, dist_comm] using hsrow) hscol end CannonAnnular end namespace CannonCapacity open Set Metric Finset CannonAnnular attribute [local instance] Classical.propDecidable lemma level_radial_hit {X : Type*} [MetricSpace X] (x : X) (K : Set X) (hK : IsPreconnected K) {b : ℝ} (hb : 0 ≤ b) (hinner : ∃ y ∈ K, dist y x ≤ b) (hdiam : 2*b < diam K) : ∃ w ∈ K, dist w x = b := by obtain ⟨y,hy,hyb⟩ := hinner obtain ⟨z,hz,hzb⟩ := exists_outer_point x (b/2) (by linarith) (by linarith : 4*(b/2) < diam K) exact hK.intermediate_value hy hz (continuous_id.dist continuous_const).continuousOn ⟨hyb,by dsimp; linarith⟩ theorem unmarked_level_detection {X : Type*} [MetricSpace X] [LocallyPathConnectedSpace X] {ι β : Type*} [Fintype ι] [Fintype β] (q : ι → X) (ξ : β → X) (h : X → X) (hh : Continuous h) (x w z : X) (A Ω K : Set X) (f : X → ℝ) (lo hi σ : ι → ℝ) (large : ι → Prop) (r s C d₀ b t : ℝ) (hr : 0 < r) (hs : 0 < s) (hrs : r ≤ s) (hC : 0 ≤ C) (hK : IsPreconnected K) (hwK : w ∈ K) (hdiam : 4*s < diam K) (hwb : dist w x = b) (hwz : dist w z ≤ s) (hregion : ∀ v, dist v w ≤ s → v ∈ A) (hdomain : ∀ v, dist v w ≤ s+r/4 → v ∈ Ω) (hlevel : ∀ v ∈ K, v ∈ Ω → f v = t) (hnet : ∀ v ∈ A, ∃ i, dist (q i) v ≤ r) (hupper : ∀ v v', dist v z ≤ 3*s → dist v' z ≤ 3*s → dist (h v) (h v') ≤ C/s*dist v v') (hlower : ∀ v v', dist v z ≤ 2*s → dist v' z ≤ 2*s → s ≤ dist v v' → d₀ ≤ dist (h v) (h v')) (hrange : ∀ i v, v ∈ Ω → dist v (q i) ≤ 2*r → f v ∈ Icc (lo i) (hi i)) (ω : β → ℝ) (hω : MacroscopicAdmissible ξ (r/s) d₀ ω) (hσ : ∀ i, 0 ≤ σ i) (hpay : ∀ i, dist (q i) z ≤ 3*s → (∑ j ∈ univ.filter (fun j => dist (ξ j) (h (q i)) ≤ (C+2)*(r/s)), ω j) ≤ σ i) (hunmarked : ∀ i, large i → |dist (q i) x-b| ≤ 5*s → t ∈ Icc (lo i) (hi i) → False) : 1 ≤ ∑ i, (Icc (lo i) (hi i)).indicator (fun _ : ℝ => if |dist (q i) x-b| ≤ 5*s ∧ ¬ large i then σ i else 0) t := by classical let V : Set X := ⋃ v ∈ K, ball v (r/4) have hV : IsOpen V := isOpen_iUnion (fun v => isOpen_iUnion (fun _ => isOpen_ball)) have hKV : K ⊆ V := by intro v hv exact mem_iUnion.mpr ⟨v,mem_iUnion.mpr ⟨hv,by simp [hr]⟩⟩ obtain ⟨v,hv,hvs⟩ := exists_outer_point w s hs.le hdiam obtain ⟨u,v',π,hπV,hu,hv',huv,hπ⟩ := annular_path_near_continuum w 0 s hs K V hK hV hKV hwK hv (by simp) (by linarith) have hzπ : ∀ a, dist (π a) z ≤ 2*s := by intro a exact (dist_triangle (π a) w z).trans (by linarith [(hπ a).2]) let π' := π.map hh have hmacro : d₀ ≤ diam (range π') := by have h₁ : dist u z ≤ 2*s := by simpa only [π.source] using hzπ 0 have h₂ : dist v' z ≤ 2*s := by simpa only [π.target] using hzπ 1 exact (hlower u v' h₁ h₂ (by linarith)).trans (dist_le_diam_of_mem (isCompact_range π'.continuous).isBounded ⟨0,π'.source⟩ ⟨1,π'.target⟩) have hadm := hω.2 (h u) (h v') π' hmacro let eligible := univ.filter (fun i => |dist (q i) x-b| ≤ 5*s ∧ ¬ large i ∧ t ∈ Icc (lo i) (hi i) ∧ dist (q i) z ≤ 3*s) let rel i := univ.filter (fun j => dist (ξ j) (h (q i)) ≤ (C+2)*(r/s)) have hcharge : ∀ j ∈ pathCellIndices ξ (r/s) π', ∃ i ∈ eligible, j ∈ rel i := by intro j hj obtain ⟨a,ha⟩ := (mem_filter.mp hj).2 obtain ⟨i,hqi⟩ := hnet (π a) (hregion _ (hπ a).2) have hiw : dist (q i) w ≤ r+s := (dist_triangle (q i) (π a) w).trans (by linarith [(hπ a).2]) have hiz : dist (q i) z ≤ 3*s := (dist_triangle (q i) w z).trans (by linarith) obtain ⟨v',hv'⟩ := mem_iUnion.mp (hπV (mem_range_self a)) obtain ⟨hv'K,hv'π⟩ := mem_iUnion.mp hv' have hv'π : dist v' (π a) < r/4 := by simpa only [mem_ball,dist_comm] using hv'π have hv'Ω : v' ∈ Ω := hdomain _ ((dist_triangle v' (π a) w).trans (by linarith [(hπ a).2])) have htv : t ∈ Icc (lo i) (hi i) := by rw [← hlevel v' hv'K hv'Ω] apply hrange i v' hv'Ω have hd := dist_triangle v' (π a) (q i) rw [dist_comm (π a) (q i)] at hd linarith have hib : |dist (q i) x-b| ≤ 5*s := by rw [← hwb] exact (abs_dist_sub_le (q i) w x).trans (by linarith) refine ⟨i,mem_filter.mpr ⟨mem_univ _,hib,fun hl => hunmarked i hl hib htv,htv,hiz⟩,?_⟩ apply mem_filter.mpr refine ⟨mem_univ _,?_⟩ have ht : dist (h (π a)) (h (q i)) ≤ C/s*r := by apply (hupper _ _ (by linarith [hzπ a]) hiz).trans exact mul_le_mul_of_nonneg_left (by simpa only [dist_comm] using hqi) (div_nonneg hC hs.le) have ha' : dist (ξ j) (h (π a)) ≤ 2*(r/s) := by simpa only [Path.map_coe,π',dist_comm,Function.comp_apply] using ha.le calc _ ≤ dist (ξ j) (h (π a)) + dist (h (π a)) (h (q i)) := dist_triangle _ _ _ _ ≤ 2*(r/s)+C/s*r := add_le_add ha' ht _ = _ := by ring have hchargeSum := finite_charge_cover eligible (pathCellIndices ξ (r/s) π') rel ω hω.1 hcharge have hbound : (∑ i ∈ eligible, ∑ j ∈ rel i, ω j) ≤ ∑ i ∈ eligible, σ i := by apply sum_le_sum intro i hi exact hpay i (mem_filter.mp hi).2.2.2.2 apply hadm.trans (hchargeSum.trans (hbound.trans _)) rw [sum_filter] apply sum_le_sum intro i _ by_cases he : |dist (q i) x-b| ≤ 5*s ∧ ¬ large i ∧ t ∈ Icc (lo i) (hi i) ∧ dist (q i) z ≤ 3*s · have hnear : |dist (q i) x-b| ≤ 5*s ∧ ¬ large i := ⟨he.1,he.2.1⟩ simp only [ite_eq_left he,indicator_of_mem he.2.2.1,ite_eq_left hnear] exact le_rfl · rw [ite_eq_right he] exact Set.indicator_nonneg (fun _ _ => by split_ifs; exact hσ i; exact le_rfl) _ end CannonCapacity end section namespace CannonCapacity open Set Metric Finset CannonAnnular CannonPacking attribute [local instance] Classical.propDecidable lemma sparse_transfer_energy {ι β : Type*} [Fintype ι] [Fintype β] (f : β → ι → ℝ) (near : β → ι → Prop) (N : ℕ) (E : ℝ) (hsupport : ∀ z i, ¬ near z i → f z i = 0) (hmult : ∀ i, (univ.filter (fun z => near z i)).card ≤ N) (he : ∀ z, ∑ i, (f z i)^2 ≤ E) : (∑ i, (∑ z, f z i)^2) ≤ (N : ℝ) * Fintype.card β * E := by classical have hrow (i : ι) : (∑ z, f z i)^2 ≤ (N : ℝ)*∑ z, (f z i)^2 := by let S := univ.filter (fun z => near z i) have heq : ∑ z, f z i = ∑ z ∈ S, f z i := by symm apply sum_filter_of_ne intro z _ hz by_contra hn exact hz (hsupport z i hn) rw [heq] have hCS := sum_mul_sq_le_sq_mul_sq S (fun _ => (1:ℝ)) (fun z => f z i) simp only [one_mul,one_pow,sum_const,nsmul_eq_mul,mul_one] at hCS calc _ ≤ (S.card : ℝ)*(∑ z ∈ S, (f z i)^2) := hCS _ ≤ (N : ℝ)*(∑ z, (f z i)^2) := by apply mul_le_mul (by exact_mod_cast hmult i) (sum_le_sum_of_subset_of_nonneg (filter_subset _ _) (fun z _ _ => sq_nonneg _)) (sum_nonneg (fun _ _ => sq_nonneg _)) (by positivity) calc _ ≤ ∑ i, (N : ℝ)*∑ z, (f z i)^2 := sum_le_sum (fun i _ => hrow i) _ = (N : ℝ)*∑ z, ∑ i, (f z i)^2 := by rw [← mul_sum,sum_comm] _ ≤ (N : ℝ)*∑ _z : β, E := mul_le_mul_of_nonneg_left (sum_le_sum (fun z _ => he z)) (by positivity) _ = _ := by simp; ring theorem exists_local_transfer_constant {X : Type*} [MetricSpace X] (Q : ℕ) (CP A c : ℝ) (hpack : PolynomialPacking (X := X) Q CP) (hCP : 0 ≤ CP) (hA : 0 < A) (hc : 0 < c) : ∃ E : ℝ, 0 ≤ E ∧ ∀ {ι β : Type*} [Fintype ι] [Fintype β] (q : ι → X) (ξ : β → X) (h : X → X) (z : X) (r s : ℝ), 0 < r → 0 < s → max A (max c 1)*(r/s) ≤ 1 → (∀ i j, i ≠ j → r ≤ dist (q i) (q j)) → (∀ i j, i ≠ j → r/s ≤ dist (ξ i) (ξ j)) → (∀ v v', dist v z ≤ 3*s → dist v' z ≤ 3*s → c/s*dist v v' ≤ dist (h v) (h v')) → ∀ ω : β → ℝ, (∑ i, (if dist (q i) z ≤ 3*s then ∑ j ∈ univ.filter (fun j => dist (ξ j) (h (q i)) ≤ A*(r/s)), ω j else 0)^2) ≤ E*∑ j, (ω j)^2 := by classical obtain ⟨kr,hkr⟩ := exists_overlap_bound Q CP A 1 hpack hCP hA zero_lt_one obtain ⟨kc,hkc⟩ := exists_overlap_bound Q CP A c hpack hCP hA hc refine ⟨(kr:ℝ)*kc,by positivity,?_⟩ intro ι β _ _ q ξ h z r s hr hs hscale hq hξ hlow ω let rel i := if dist (q i) z ≤ 3*s then univ.filter (fun j => dist (ξ j) (h (q i)) ≤ A*(r/s)) else ∅ let S := univ.filter (fun i => dist (q i) z ≤ 3*s) have hratio : 0 < r/s := div_pos hr hs have hscale₁ : max A 1*(r/s) ≤ 1 := (mul_le_mul_of_nonneg_right (max_le_max le_rfl (le_max_right ..)) hratio.le).trans hscale have hscalec : max A c*(r/s) ≤ 1 := (mul_le_mul_of_nonneg_right (max_le_max le_rfl (le_max_left ..)) hratio.le).trans hscale have hrow (i : ι) : (rel i).card ≤ kr := by dsimp only [rel] split_ifs · have hh := hkr ξ univ (h (q i)) (r/s) hratio hscale₁ (by simpa using hξ) simpa only [dist_comm] using hh · simp have hcol (j : β) : (univ.filter (fun i => j ∈ rel i)).card ≤ kc := by have him : ∀ i ∈ S, ∀ j ∈ S, i ≠ j → c*(r/s) ≤ dist (h (q i)) (h (q j)) := by intro i hi j hj hij calc _ = c/s*r := by ring _ ≤ c/s*dist (q i) (q j) := mul_le_mul_of_nonneg_left (hq i j hij) (div_nonneg hc.le hs.le) _ ≤ _ := hlow _ _ (mem_filter.mp hi).2 (mem_filter.mp hj).2 have hh := hkc (fun i => h (q i)) S (ξ j) (r/s) hratio hscalec him convert hh using 1 congr 1 ext i by_cases hi : dist (q i) z ≤ 3*s <;> simp [S,rel,hi] have he := overlap_energy rel ω kr kc hrow hcol convert he using 1 congr 1 funext i simp only [rel] split_ifs <;> simp end CannonCapacity end namespace CannonCapacity open Set Metric Finset CannonAnnular attribute [local instance] Classical.propDecidable theorem node_separated_capacity {X : Type*} [MetricSpace X] [LocallyPathConnectedSpace X] {ι β ζ δ : Type*} [Fintype ι] [Fintype β] [Fintype ζ] [Fintype δ] (q : ι → X) (bin : β → ℝ) (aux : ζ → X) (ξ : δ → X) (h : ζ → X ≃ₜ X) (x : X) (Ω : Set X) (f : X → ℝ) (K : ℝ → Set X) (lo hi t : ι → ℝ) (a b A B₀ R r s C c d₀ H L Q E M BinMul : ℝ) (N Naux : ℕ) (hab : a < b) (hA : 1 ≤ A) (hAB : A+2 ≤ B₀) (hR : 0 < R) (hr : 0 < r) (hs : 0 < s) (hrs : r ≤ s) (hsR : s ≤ R/4) (hC : 0 ≤ C) (hc : 0 < c) (hdc : d₀ ≤ c) (hbin : ∀ j, (A+1)*R ≤ bin j ∧ bin j ≤ (B₀-1)*R) (hnet : ∀ v, (A-1)*R ≤ dist v x → dist v x ≤ (B₀+1)*R → ∃ i, dist (q i) v ≤ r) (haux : ∀ v, (A-1)*R ≤ dist v x → dist v x ≤ (B₀+1)*R → ∃ z, dist v (aux z) ≤ s) (hΩ : ∀ v, dist v x ≤ (B₀+2)*R → v ∈ Ω) (hK : ∀ v ∈ Ioo a b, IsPreconnected (K v)) (hinner : ∀ v ∈ Ioo a b, ∃ y ∈ K v, dist y x ≤ A*R) (hdiam : ∀ v ∈ Ioo a b, 2*B₀*R < diam (K v)) (hlevel : ∀ v ∈ Ioo a b, ∀ y ∈ K v, y ∈ Ω → f y = v) (hlohi : ∀ i, lo i ≤ hi i) (ht : ∀ i, 0 ≤ t i) (hlen : ∀ i, clip a b (hi i)-clip a b (lo i) = (b-a)*t i) (hrange : ∀ i v, v ∈ Ω → dist v (q i) ≤ 2*r → f v ∈ Icc (lo i) (hi i)) (hupper : ∀ z v v', dist v (aux z) ≤ 3*s → dist v' (aux z) ≤ 3*s → dist (h z v) (h z v') ≤ C/s*dist v v') (hlower : ∀ z v v', dist v (aux z) ≤ 3*s → dist v' (aux z) ≤ 3*s → c/s*dist v v' ≤ dist (h z v) (h z v')) (ω : δ → ℝ) (hω : MacroscopicAdmissible ξ (r/s) d₀ ω) (hωmass : ∑ j, (ω j)^2 ≤ M) (hlocal : ∀ z, (∑ i, (if dist (q i) (aux z) ≤ 3*s then ∑ j ∈ univ.filter (fun j => dist (ξ j) (h z (q i)) ≤ (C+2)*(r/s)), ω j else 0)^2) ≤ E*∑ j, (ω j)^2) (hmultaux : ∀ i, (univ.filter (fun z => dist (q i) (aux z) ≤ 3*s)).card ≤ Naux) (hL : 0 < L) (hQ : 0 < Q) (hE : 0 ≤ E) (hM : 0 ≤ M) (hBinMul : 0 < BinMul) (hHr : 0 ≤ H*r) (hN : 0 < N) (hdegree : ∀ i, (univ.filter (fun j => dist (q i) (q j) ≤ H*r)).card ≤ N) (hmult : ∀ i, ((univ.filter (fun j => |dist (q i) x-bin j| ≤ 5*s)).card : ℝ) ≤ BinMul) (hlarge : 2*BinMul*N ≤ L*Fintype.card β) (hsmall : 4*BinMul^2*((Naux:ℝ)*Fintype.card ζ*(E*M))*N < Q*(Fintype.card β : ℝ)^2) : ∃ S : Finset ι, (∀ i ∈ S, ∀ j ∈ S, i ≠ j → H*r < dist (q i) (q j)) ∧ 1 ≤ ∑ i ∈ S, min (L*t i) (Q*(t i)^2) := by classical let σz : ζ → ι → ℝ := fun z i => if dist (q i) (aux z) ≤ 3*s then ∑ j ∈ univ.filter (fun j => dist (ξ j) (h z (q i)) ≤ (C+2)*(r/s)), ω j else 0 let σ : ι → ℝ := fun i => ∑ z, σz z i have hσz : ∀ z i, 0 ≤ σz z i := by intro z i dsimp only [σz] split_ifs · exact sum_nonneg (fun j _ => hω.1 j) · exact le_rfl have hσ : ∀ i, 0 ≤ σ i := fun i => sum_nonneg (fun z _ => hσz z i) have he := sparse_transfer_energy σz (fun z i => dist (q i) (aux z) ≤ 3*s) Naux (E*M) (by intro z i hn; simp only [σz,ite_eq_right hn]) hmultaux (fun z => (hlocal z).trans (mul_le_mul_of_nonneg_left hωmass hE)) apply separated_capacity_of_detection q lo hi t σ (fun j i => |dist (q i) x-bin j| ≤ 5*s) a b BinMul L Q ((Naux:ℝ)*Fintype.card ζ*(E*M)) H r N hab hlohi ht hlen hσ hL hQ hBinMul (by positivity) hHr hN hdegree hmult he hlarge hsmall intro j v hv hunmarked have hbnon : 0 ≤ bin j := by nlinarith only [(hbin j).1,hA,hR] obtain ⟨w,hwK,hwb⟩ := level_radial_hit x (K v) (hK v hv) hbnon (by obtain ⟨y,hy,hyA⟩ := hinner v hv exact ⟨y,hy,by nlinarith only [(hbin j).1,hyA,hR]⟩) (by nlinarith only [hdiam v hv,(hbin j).2,hR]) have hwann : (A-1)*R ≤ dist w x ∧ dist w x ≤ (B₀+1)*R := by rw [hwb] constructor <;> nlinarith only [(hbin j).1,(hbin j).2,hR] obtain ⟨z,hwz⟩ := haux w hwann.1 hwann.2 apply unmarked_level_detection q ξ (h z) (h z).continuous x w (aux z) {v | (A-1)*R ≤ dist v x ∧ dist v x ≤ (B₀+1)*R} Ω (K v) f lo hi σ (fun i => L ≤ Q*t i) r s C d₀ (bin j) v hr hs hrs hC (hK v hv) hwK (by nlinarith only [hdiam v hv,hAB,hA,hR,hsR]) hwb hwz (by intro y hy have hyw := dist_triangle y w x have hwy := dist_triangle w y x rw [dist_comm w y] at hwy constructor <;> nlinarith only [(hbin j).1,(hbin j).2,hy,hyw,hwy,hwb,hsR,hR]) (by intro y hy apply hΩ have hh := dist_triangle y w x nlinarith only [(hbin j).2,hh,hy,hwb,hsR,hrs,hR]) (hlevel v hv) (by intro y hy; exact hnet y hy.1 hy.2) (hupper z) (by intro y y' hy hy' hyy' have hh := hlower z y y' (by linarith only [hy,hs]) (by linarith only [hy',hs]) have hsne : s ≠ 0 := ne_of_gt hs have hcs : c/s*s = c := by field_simp have hm := mul_le_mul_of_nonneg_left hyy' (div_nonneg hc.le hs.le) rw [hcs] at hm exact hdc.trans (hm.trans hh)) hrange ω hω hσ (by intro i hi have hh : σz z i ≤ σ i := single_le_sum (fun z _ => hσz z i) (mem_univ z) simpa only [σz,ite_eq_left hi] using hh) hunmarked end CannonCapacity end section namespace CannonCapacity open Set Metric Finset CannonAnnular CannonPacking attribute [local instance] Classical.propDecidable end CannonCapacity end end end end end end end end end end end end end end end OAI